A numerical method for computing hydrometallurgical activity-activity diagrams

HydrometaUurgy, 4 (1979) 217--232

217

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

A NUMERICAL METHOD FOR COMPUTING HYDROMETALLURGICAL ACTIVITY--ACTIVITY DIAGRAMS

K. O S S E O - A S A R E

Metallurgy Section, Department of MaterialsScience and Engineering, The Pennsylvania State University, UniversityPark, P A 16802 (U.S.A.) and T H O M A S

H. B R O W N

Department of Geological Sciences, Universityof British Columbia, Vancouver, B.C. V 6 T 1 W 5 (Canada) (Received July 24th, 1978; accepted January 10th, 1979)

ABSTRACT Osseo-Asare, K. and Brown, T.H., 1979. A numerical method for computing hydrometallurgicalactivity--activitydiagrams. Hydrometallurgy, 4: 217--232. A modification of the DIAGRAM computer program has been developed to facilitate graphical representation of hydrometallurgical equilibria. The method is based on dissociation equations with corresponding log equilibrium constants, Ki, and log reaction quotients, Qi; Ki equals Qi at equilibrium. The ith dissociation equation is written in terms of Qi, the log activity of the ith metallic species (Pi), and a certain set of variables (Vi), representing the log activities of species whose stability fields are not to be plotted (e.g., log{Me 2+ }, logIe }, Iog{H+}): n

Qi ~ BiPi + ~

CaV)

1=I where B i is the reaction coefficient of the ith metallic species in the ith dissociation equation, and Cii is the reaction coefficient of the jth variable in the ith dissociation equation. One of the variables is designated the balancing variable and allows comparison of the relative stability of any two metallic species to be made in terms of a single equation. By means of the balancing variable, the program generates internally the re(m--1 )/2 relative stability equations linking pairs of the m metal-containing species. The stability region of each metallic species is then determined by a systematic scanning of the plotting area using the criterion (KL > QL?). DIAGRAM can calculate and plot stability diagrams using any two of the system variables. Thus not only Eh-pH diagrams, but additional plots of log {Metal}--pH, log{Metal}--Eh, log{NH3 + NH4+}--pH, etc. can be generated readily. The numerical and thermodynamic bases of the program are described and the capability of DIAGRAMis illustrated with some selected hydrometallurgical examples.

INTRODUCTION Diagrams which delineate t h e relative chemical stabilities o f various species present in the same aqueous environment are o f considerable aid in under-

218 standing and predicting equilibrium reactions in hydrometallurgy (Peters, 1976; Robins, 1968). The most well-known of such stability diagrams is the Eh-pH diagram made popular by the work of Pourbaix (1949, 1966), Garrels (1960), and Garrels and Christ (1965). Eh--pH diagrams enable one to see at a glace the range of stability of aqueous species, as well as solid phases, under specific conditions of Eh (or oxidation potential) and pH. In order to determine the stability fields for m species, m ( m - 1 ) / 2 equations must be generated relating all possible pairs of species. As the number of species increases, thc ~ecessary calculations become quite time consuming. For example, a system with 15 species will requite 105 equations. Thus, there is growing interest in the use of the c o m p u t e r to facilitate the otherwise tedious calculations (Brook, 1971; Froning et al., 1976; Duby, 1976; Verhulst and Duby, 1977). The available published information indicates that most of the effort has gone into developing c o m p u t e r programs to calculate Eh--pH diagrams. Yet, as discussed elsewhere (Osseo-Asare and Fuerstenau, 1978; OsseoAsare, 1975), Eh--pH diagrams constitute only one example of a general class of activity--activity diagrams. In hydrometallurgical systems, in addition to Eh and pH, it is also useful to consider plotting with additional coordinate pairs such as log {Me } --pH, log (NHa + NH4 ÷ } --pH etc.; activity ratio diagrams can also be instructive (Helgeson et al., 1969). A general c o m p u t e r program, DIAGRAM, is n o w available which permits rapid calculation and plotting of species predominance fields in terms of log(activity)--log(activity) diagrams. Unlike previous c o m p u t e r programs (Brook, 1971; Froning et al., 1976; Duby, 1976; Verhulst and Duby, 1977), DIAGRAM is n o t limited to Eh-pH plots; users have the option of specifying their own pairs of plotting coordinates. DIAGRAM was originally developed for geochemical systems by Brown (1970). A preliminary modification of DIAGRAM was subsequently made by Osseo-Asare (1975) to suit the needs of hydrometallurgical problems. This communication discusses the thermodynamic and numerical bases of an updated version of this earlier modification of DIAGRAM and presents some examples to illustrate its use in hydrometallurgy. METHOD

Dissociation equations Application of DIAGRAM requires that a careful distinction be made between phases and variables. In this context, phase refers to any c o m p o u n d (irrespective of whether it is a dissolved aqueous species or a more conventional phase such as a solid) whose stability field is under consideration. A dissociation equation must be written for each phase in terms of a certain set of compounds or variables whose stability fields are not to be plotted. In a strict thermodynamic sense, the variables are one of the possible c o m p o n e n t sets for the system. The dissociation equation is written with the phase and all the variables on the right hand side; the reaction coefficient of each phase and variable

219

must be specified. The reaction coefficient of a phase or a variable is the number of molecules of that c o m p o u n d which participates in a given dissociation reaction. One of the variables is used to balance the equations relating pairs of phases. The balancing variable must either contain an element which is c o m m o n to all the phases, or it must appear with a non-zero reaction coefficient in all the phase dissociation reactions. T w o of the remaining variables serve as the plotting coordinates; the rest of the variables must be assigned constant values. As an illustration, consider the stability relations between the c o m p o u n d s Cu°--Cu20--CuO. The corresponding dissociation reactions, written in the conventional manner are of the form, Cu °= Cu 2÷ + 2e

Q'I, K'~

Cu20 + 2H ÷ = 2Cu :÷ + 2e + H20

Q2, g~

CuO + 2H ÷ = Cu2+ + H20

(1)

t

(2)

r

(3)

Q~ K3

where Q~ and K~ are, respectively, the reaction quotient and equilibrium constant for the ith dissociation equation. At equilibrium, Ki' = Q~. For any arbitrary solution conditions, the reaction quotients are given by Q'I = (Cu 2+} (e~2/(Cu °)

(4)

Q~ = (Cu2+) 2 {e) 2 ( H ~ O ) / ( C u 2 0 ) (H+) 2

(5)

Q; = (Cu 2÷) ( H 2 0 } / { C u O ) (H÷} 2

(6)

The DIAGRAM forms of equations (1) to (6) are, log Q'I = - 1 log(Cu ° } + l l o g ( C u 2÷) + 2 l o g ( e ) + 0 l o g ( H ÷ ) + 0 1 o g ( H 2 0 )

(7)

logQ~ = - 1 log{Cu20 } + 2 log(Cu2+) + 2 l o g ( e ) - 2 log(H÷) + l l o g ( H 2 0 }

(8)

logQ~ = - 1 log(CuO} + 1 log(Cu2÷} + 0 l o g ( e } - 2 l o g ( H ÷ ) + 1 l o g ( H 2 0 )

(9)

In these equations, Cu °, Cu20 and CuO constitute the phases. The variables are (the logs of) Cu 2÷, e, H ÷, and H20; the corresponding reaction coefficients are 1, 2, 0, 0 in equation (7); 2, 2, - 2 , 1 in equation (8); and 1, 0, - 2 , 1 in equation (9).

Balancing The balancing variable allows comparison of the relative stability of two phases to be determined in terms of a single equation. In reactions (7)--(9), balancing can be accomplished with Cu 2÷ to give the following equations,

220

Cu°/Cu20: 2 log QI -log Q~

2 C u ° = 2 C u 2+ + 4e 2 C u 2+ + 2e + H 2 0 = C u 2 0 + 2 H + 2Cu ° + H20

= Cu20

+ 2e +

l o g Q 4' = 2 l o g Q l' - l o g Q2

2H ÷

(10)

Cu°/CuO:

log QI - - l o g Q;

Cu ° = Cu ~÷ + 2e Cu 2÷ + H 2 0 = C u O + 2 H +

log Qs = log Q i - l o g Q~

Cu ° + H 2 0 = C u O + 2e + 2 H +

(il)

Cu20/CuO: C u 2 0 + 2 H ÷= 2Cu 2+ + 2e + H 2 0

log Q~

2Cu 2÷ + 2 H 2 0 = 2 C u O + 4 H ÷

- 2 log Q~

C u 2 0 + H 2 0 = 2 C u O + 2e + 2 H ÷

•

log Q'6= log Q :' - 2

1

og Q3'

(12)

Since at e q u i l i b r i u m log K~= log Q~, e q u a t i o n s ( 1 0 ) - ( 1 2 ) c a n be r e w r i t t e n in t e r m s o f t h e f o l l o w i n g p h a s e b o u n d a r y e q u a t i o n s : Cu°/Cu20:

pe = - l h log K ~ - p H

Cu°/CuO:

pe = - l h log K s ' - p H

Cu20/CuO:

pe = -1/i log K ~ - p H

w h e r e pe = - l o g { e } Thus, b a l a n c i n g on Cu 2÷ allows t h e p l o t t i n g o f p e - - p H d i a g r a m s w h i c h s h o w t h e relative stabilities o f Cu ° , C u 2 0 a n d CuO. H o w e v e r , if in a d d i t i o n t o Cu °, C u 2 0 a n d CuO, it is also desired t o investigate t h e s t a b i l i t y field o f Cu 2÷, t h e n this species m u s t b e t r e a t e d b o t h as a p h a s e a n d as a variable. A m e a n s o f acc o m p l i s h i n g this is t o i n c l u d e w i t h e q u a t i o n s (1) t o (3), a d i s s o c i a t i o n e q u a t i o n f o r Cu 2÷ as a phase, w r i t t e n in t e r m s o f Cu 2÷ as a variable: Cu 2÷ = Cu 2+

K '= 1

(13)

Alternatively, the required balancing may be achieved by postulating a hypot h e t i c a l a q u e o u s species Cu 3+ s u c h t h a t t h e r e is a zero free e n e r g y c h a n g e (i.e., an e q u i l i b r i u m c o n s t a n t o f u n i t y ) f o r t h e r e a c t i o n : Cu 2* = Cu 3+ + e

K0 = 1

T h e r e a c t i o n s (1) t o (3) n o w b e c o m e e q u i v a l e n t to: Cu °

= Cu 3÷ + 3e

K~ '= K;K~)

221 rr

r

t2

Cu20 + 2H + = 2Cu 3÷ + 4e + H~O

K2 = K2 (K0)

C u O + 2H ÷ = Cu 3÷ + e + H20

K3'

C u ~+ = C u 3+ + e

K ; ' = Ko

= K3Ko ' '

or in DIAGRAM format, logQ~'=-llog(Cu°)+llog(Cu3+)+31og(e)

+ 01og(H +) + 0 l o g ( H 2 0 )

log Q~'=- l l o g ( C u 2 0 ) + 2 log (Cu a+ ) + 4 l o g ( e ) - 2 log(H +) + 1 l o g ( H 2 0 ) fl

(14) (15)

log Q3 - - 1 log (CuO) + 1 log (Cu 3+) + 1 log (e) ~ 2 log (H + ) + 1 log ( H 2 0 )

(16)

log Q ~ ' = - 1 log(Cu 2+) + 1 log(Cu 3+) + 1 log(e) + 0 log(H +) + 0 l o g ( H 2 0 )

(17)

Matrix n o t a t i o n

The above equations can be written in matrix form b y making the following substitutions. Let log(Cu a+ ) be the first variable, log(e ) t h e second, l o g ( H + ) the third, and log ( H 2 0 ) the fourth. That is, if the symbol Vi is used to represent the ith variable, then V1 = log(Cu 3+) 172 = log ( e ) V3 = l o g ( H + ) V4 = l o g ( H 2 0 ) Furthermore, let Cij = reaction coefficient of the jth variable in the ith dissociation equation, Bi = reaction coefficient of the phase in the ith dissociation equation, Q i = the log of the reaction quotient for the ith dissociation equation, Ki = the log of the equilibrium constant for the ith dissociation equation, Pi = the log of the activity of the ith phase, then equations (14)--(17) become: Q, = B , P I + Cll V1 + C,2V2 + C,3V3 + C,4V4 Q2 = B2P2 + C2,V, + C22V2 + C23V3

+

C24V4

Q3 = B3P3 + C31V1 + C32V2 + C33V3 + C34V4 Q4 = B4P4 + C41V1 + C42V2 + C43V3 + C44V4

or generalizing and using matrix notation for a system consisting of m phases and n variables, Q=S+CV At equilibrium,

K = S* + C V*

(18)

where C is the m× n matrix (i.e., m rows and n columns) with elements Ci],

222 V is a column vector with n rows, and Q, K and S are each column vectors with m rows:

Q1

+2

V1

Q=

v

S1

K1 K2 i

K=

i

S=

L L +

Si +

i i

v.

K,

sm

and

Si = BiPi The asterisks in equation (18) denote the equilibrium values of S and V. In order to use DIAGRAM, the elements in the vectors V and K, and the C matrix must be supplied as input data. For simplicity, all the dissociation reactions are written for 1 mole of the phase, so that all the B i's have a value of - 1. For solid phases, unit activity is assumed. Thus Pi = log (activity) = 0 for all solids, so that Si = BiPi = 0 for all solid phases. In the case of the aqueous "phases", the Pi values (which should be constants) must be supplied in the input data. DISCUSSION DIAGRAM was originally written for geochemical systems with primary emphasis on the stability regions of solid phases. Therefore, the starting dissociation equations were of the form: n

g i = ~ CijVj j=l We shall hereafter use DIAGRAM@ to represent this earlier version of DIAGRAM. Application of DIAGRAM@ to aqueous "phases" is limited to the case where the aqueous phases are present at a constant activity of unity. Examples of the plotted diagrams using DIAGRAM@ have been compiled by Helgeson, Brown, and Leeper (1969). In order to m o d i f y DIAGRAM@ to suit hydrometallurgical problems, OsseoAsare (1975) introduced the S vector: n

g i = si + ~ Cij Vj j=l where

Si = BiPi We shall designate this modified version of DIAGRAM as DIAGRAM1. Figure 1 presents a simplified flowchart for DIAGRAM1.

223

f [

MAIN Read data: V, K, C, P

I "

4___. PHASEQ Generate m(m-l)/2 relative phase stability equations

.

I

A r r a n g e t h e c o l u m n s o f t h e CV M a t r i x in t h e o r d e r o f t h e p l o t t i n g ~ g a r i a b l e s The X a x i s v a r i a b l e in the first column The Y a x i s v a r i a b l e in the second column

1.

2. 3.

STA~REC Divide plotting area grid points.

into

60x60

Determine the stable phase at each grid point. Determine the phase boundaries

r- . . . . . . . . . . . . . . . . .

~

~ :,B

Find center x~y coordinates of each stability field I I. 2.

/

PLOTEQ Plot Stability Lines Print centered Labels on stability fields

Print XY coordinates of Stability line intercepts

Fig.1. Flowchart for DIAGRAM1.

At present, D I A G R A M 1 m a y be used with two versions of the S T A R E C subroutine: S T A R E C 1 and S T A R E C 2 . S T A R E C 1 allows all pairs of plotting coordinates except those involving the balancing variable. Thus, in hydrometallurgical systems, where a metal-containing variable is used for balancing, S T A R E C 1 can be used to generate diagrams such as Eh--pH, log{NH4 + + NH3) --pH, log {NH4 + + N H 3 )--Eh, etc. (Osseo-Asare and Fuerstenau, 1978; OsseoAsare, 1975). S T A R E C 2 is useful when it is desired to use the log(activity) of the balancing element as a plotting variable as well. For example, S T A R E C 2 allows one to balance with Cu 3+ and stillbe able to generate log(Cu}--pH diagrams, where (Cu } represents the total concentration of dissolved copper species. The difference between the two versions is that in S T A R E C I , the S' vector is a constant for a given plot, whereas in S T A R E C 2 , S' varies with log{Cu ) (assuming, for example, that log{Cu } is the y-axis).

224 For an illustration of the use of DIAGRAM1, let us consider the Cu--H20 and Cu--NH3--H20 systems. The relevant thermodynamic data needed for the application of the computer program are presented in Table 1. The dissociation constants are based on the compilations of Wagman et al. ( 1 9 6 6 - - 1 9 7 1 ) and Sillen and Martell ( 1 9 6 4 - - 1 9 7 1 ) . In addition to the equilibrium data, the physical size of the plot, the extreme values of the plotting variables, as well as the t w o species which will serve as the plotting variables must be specified. The activities of the non-plotting variables must be given constant values. In DIAGRAM1 terminology, the C u - - H 2 0 system involves ten phases, i.e., the first ten species in Table 1. The variables in the dissociation equations are Cu 3÷, e, H ÷ and H20. The activity of water may be taken as unity so that the first three variables can be used to calculate diagrams of Eh--pH, log{Cu }--pH, and log{Cu }--Eh as shown in Figs.2--4, respectively. Figure 2 is an example of what is sometimes called a Pourbaix diagram; it delineates the Eh and pH ranges which favor copper dissolution from Cu °, Cu20 and CuO. The effect of dissolved metal concentration is usually shown Table 1

Thermodynamic data for the Cu-H20 and Cu-NH3-H20 systems at 25°C

Species Conventional _. symbol

Computer symbol

Cu ° Cu_O Cu~ Cu(OH)o Cu 2+ ~ Cu+ Cu(OH) + Cu(OH)2° CuO 2 -

llCuO~

Cu Cu20 CuO Cu(OH)2 Cu++ Cu+ Cu(OH)+ Cu(OH)2, AQ HCuO2Cu02--

Cu(NHR) + Cu (NH~) ?+

CuA+ CuA2+

Cu ( ~ )

2~,

Nature of uncomplexed ammonia

NH4+ or NH 3

CuA'~

Cu (NII~)o"~ Cu(NH~)~2+ Cu(NH~)~ 2+ Cu(NN~)~2+

CuA24-F CuA3q-F CuA4++ CuAS++

Cu(NH~) + Cu(NH~)2+ Cu(~o)2+ Cu(NH~)22+ Cu(~I~)32+ Cu(NH~)~ 2+ Cu(NHj)~2+

CuA+ CuA2+ CuA++ CuA2~-F CuA3q-~ CuA4++ CuA5~-~

NH4 +

~I 3

Reaction coefficient

Dissociation constant

Cu 3+

e

H+

NH4+/NH 3

H20

log K

i 2 i i 1 i i i I i

3 4 1 i 1 2 1 i i l

0 -2 -2 -2 0 0 -i -2 -3 -4

0 0 0 0 0 0 0 0 0 0

0 i i 2 0 0 i 2 2 2

-11.48 -7.00 7.34 9.08 0. -2.72 7.48 14.23 27.39 40.78

1 1

2 2

-i -2

1 2

0 0

0.75 5.22

I

1

-i

i

0

5.06

i i I i

i 1 i I

-2 -3 -4 -5

2 3 4 5

0 0 0 0

I0.76 17.06 24.12 33.87

i 1 i I l l 1

2 2 i i i i i

0 0 0 0 0 0 0

i 2 i 2 3 4 5

0 0 0 0 0 0 0

-8.50 -13.28 -4.19 -7.74 -10.69 -12.88 -12.38

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on Eh--pH diagrams by means of additional stability lines corresponding to each metal ion concentration. Often, the result is a tangle o f lines which is hard to decipher. The use of log (Metal }--pH diagrams can provide an alternative to the superimposed Eh--pH diagram approach. In the study of the solution equilibria of leaching processes, log (Metal }--pH diagrams are helpful in estimating the solubility limits of the dissolving metals. Thus, in Fig.3, the Cu°/Cu 2÷ stability boundary in acidic solutions is in agreement with the finding that dissolution of cuprite, Cu20, in oxygen-starved solutions yields metallic copper and cupric ions (Wadsworth and Wadia, 1955). Figure 4 is a log (Cu }--Eh diagram for constant pH of 8.5. Figure 5, which shows a pH-Eh diagram, demonstrates the flexibility of the DIAGRAM1 computer program. There are several important hydrometallurgical systems in which, in addition to the metallic species, the equilibria of nonmetallic species (e.g., NH3--H20 and S--H20) also need to be incorporated into the phase stability calculations. Application of DIAGRAM1 to such systems can take one of two possible forms. The simplest approach is to make separate runs of the program for the solution conditions under which a given nonmetallic compound is the predominant species. A composite diagram can then be developed from the contributions of the partial plots. Alternatively, an additional subroutine can be incorporated into DIAGRAM1 to alter internally the reaction coefficients and the dissocia-

227

tion equation equilibrium constants as the plotting space is traversed. The phase diagrams for the Cu--NH3--H20 system presented in Figs.6--11, illustrate the capability of DIAGRAM1 to treat complex aqueous systems. According to Table 1, the Cu--NH3--H20 system involves seventeen phases and five variables. In terms of the independent variables Cu 3., e, H ÷ and (NH4 + NH3), at least six types of diagrams can be generated: Eh--pH, log {Cu ~-pH, log{Cu ) --Eh, Eh--log{NH4 + NH3}, log {Cu }--log {NH4 + NH3}, and log(NH4 + NH3) --pH. Figures 6--11 present in a vivid manner, the chemical basis of ammonia leaching i.e., the presence of the metal--ammine stability regions (Osseo-Asare and Fuerstenau, 1978; Osseo-Asare, 1975; Han et al., 1974). The following equilibrium exists between NH4 ÷ and NH3: NH4+ ~ NH3 + H ÷ logK = --9.25 at 25°C Thus when the pH is less than 9.25, the predominant form of aqueous ammonia is NH4* while NH3 is the predominant ammonia species when the pH exceeds 9.25. Accordingly, diagrams plotted with pH as a variable require two sets of dissociation equations, involving NH4* and NH3 respectively, in the low Q i

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CUO

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D

D

CU++ I

-2. D

-1.~

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0.0

D.~

1.0

i

1. F

2.0

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231

and high pH regions. As can be seen from Table 1, in the case of the c o p p e r ammines, the reaction coefficients for H ÷ have non-zero values in the NH4 ÷ region whereas in the NH3 region, the H ÷ reaction coefficients have zero values. In Figures 6, 7 and 11, the composite diagrams were generated in one job. Only the NH4 ÷ region data were used as input to DIAGRAM1. After completion of the NH4 ÷ portion of the calculation, the reaction coefficients and dissociation constants corresponding to the NH3 region were generated internally and the calculation repeated. The coordinates of the stability fields in the NH4 ÷ and NH3 regions were then supplied simultaneously to PLOTEQ, the plotting subroutine. The Cu~O/Cu(NH3): ÷ boundary in Fig.9 illustrates the fact that line segments which are less than 1/60th of the axis length are n o t plotted by DIAGRAM. However, the position of the lines are obvious and they can be added by hand; alternatively, the grid size may be increased. DIAGRAM1 is written in F O R T R A N IV. Entry, manipulation of records as well as plotting have all been performed interactively from a terminal. However, card decks may also be used. At present, subroutine PLOTEQ is based on the Quick Draw Graphics System (QDGS) of the Pennsylvania State University Computation Center. However, conversion to other graphics systems can be readily accomplished. Computer facilities without plotting capabilities can still utilize the DIAGRAM1 program since the coordinates of the stability boundaries are also provided as output. ACKNOWLEDGEMENTS

K.O.A.'s modification of DIAGRAMO began while he was a Jane Lewis Fellow in the Materials Science and Engineering Department of the University of California, Berkeley. He gratefully acknowledges the encouragement of Professors D.W. Fuerstenau and H.C. Helgeson. The work has continued at The Pennsylvania State University with the help of a Research Initiation Grant and a Faculty Research Grant. DIAGRAM~ was originally developed at the Vogelback Computing Centre at Northwestern University when T.H.B. was a student of Professor H.C. Helgeson and his help is gratefully acknowledged. REFERENCES* Brook, P.A., 1971. A computer method of calculating potential--pH diagrams. Corros. Sci., 11: 389--396. Brown, T.H., 1970. Theoretical Prediction of Equilibria and Mass Transfer in the System CaO--MgO--SiO2--H20--CO2--NaCl--HC1. Ph.D. Thesis, Northwestern University, Evanston, Illinois. Duby, P., 1976. Graphical representation of equilibria in aqueous systems at elevated temperatures. In: R.W. Staehle, D. de G. Jones and J.E. Slater (Eds.), High Temperature High Pressure Electrochemistry in Aqueous Solutions. National A s s o c i a t i o n o f Corrosion Engineers. *A copy of the DIAGRAM1 program (cards, tape or printout) can be o b t a i n e d from K.O.A. for the cost of handling and postage.

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Froning, M.H., Shanley, M.E. and Verink, Jr., E.D., 1976. A n improved method for calculating of potential--pH diagrams of metal--ion--water systems by computer. Corros. Sci., 16: 371--377. Garrels, R.M., 1960. Mineral Equilibria, Harper and Row, N e w York, 254 pp. Garrels, R.M. and Christ, C.L., 1965. Solutions, Minerals and Equilibria, Harper and Row, N e w York, 450 pp. Han, K.N., Hoover, M. and Fuerstenau, D.W., 1974. A m m o n i a - - a m m o n i u m leaching of deep-sea manganese nodules. Int. J. Min. Process., 1: 215--230. Helgeson, H.C., Brown, T.H. and Leeper, R.H., 1969. Handbook of Theoretical Activity Diagrams Depicting Chemical Equilibria in Geologic Systems Involving an Aqueous Phase at One A t m and 0o--300 ° C. Freeman, Cooper & Co., San Francisco. Osseo-Asare, K., 1975. A m m o n i a Hydrometallurgy: Thermodynamics of Dissolution and Adsorption Phenomena in Oxide Systems. Ph.D. Thesis, University of California, Berkeley. Osseo-Asare, K. and Fuerstenau, D.W., 1978. Application of activity--activitydiagrams to ammonia hydrometallurgy. The systems Cu--NHs--H20 , Ni--NH3--H20, and Co--NH3--H20 at 25 ° C. In: T.W. Chapman, L.L. Tavlarides, G.L. Hubred and R.M. Wellek (Eds.), Fundamental Aspects of Hydrometallurgical Processes. A I C h E Symposium Series, 74 (173): 1--13. Peters, E., 1976. Direct leaching of sulfides: chemistry and application. Met. Trans. B, 7B: 505--517. Pourbaix, M., 1949. Thermodynamics of Dilute Aqueous Solutions,~Edward Arnold and Co., London, 136 pp. Pourbaix, M., 1966. Atlas of Electrochemical Equilibria in Aqueous Solutions, Pergamon, London, 644 pp. Robins, R.G., 1968. The Application of Potential--pH Diagrams to the Prediction of Reactions in Pressure Hydrothermal Processes. Warren Spring Laboratory, L R 90 (MSF), Stevenage. Sillen, L.G. and Martell, A.E., 1964--1971. Stability Constants of Metal--lon Complexes. Special Publication No., 17, No. 25, The Chemical Society, London. Verhulst, D. and Duby, P., 1977. A computer program for calculating and plotting potential-p H diagrams. In: P. Duby, The Thermodynamic Properties of Aqueous Inorganic Copper Systems. International Copper Research Association. Wadsworth, M.E. and Wadia, D.R., 1955. Reaction rate study of the dissolution of cuprite in sulfuricacid. Trans. AIME, 209: 755--759. Wagman, D.D. et al.,1966--1971. Selected Values of Chemical Thermodynamic Properties. N B S Technical Note 270, parts 1--6, NBS, Washington.